Integrand size = 15, antiderivative size = 43 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {2 x^3}{7 a \left (a+b \sqrt {x}\right )^7}+\frac {x^3}{21 a^2 \left (a+b \sqrt {x}\right )^6} \]
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.77 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx=\frac {-a^5-7 a^4 b \sqrt {x}-21 a^3 b^2 x-35 a^2 b^3 x^{3/2}-35 a b^4 x^2-21 b^5 x^{5/2}}{21 b^6 \left (a+b \sqrt {x}\right )^7} \]
(-a^5 - 7*a^4*b*Sqrt[x] - 21*a^3*b^2*x - 35*a^2*b^3*x^(3/2) - 35*a*b^4*x^2 - 21*b^5*x^(5/2))/(21*b^6*(a + b*Sqrt[x])^7)
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{5/2}}{\left (a+b \sqrt {x}\right )^8}d\sqrt {x}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle 2 \left (\frac {\int \frac {x^{5/2}}{\left (a+b \sqrt {x}\right )^7}d\sqrt {x}}{7 a}+\frac {x^3}{7 a \left (a+b \sqrt {x}\right )^7}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle 2 \left (\frac {x^3}{42 a^2 \left (a+b \sqrt {x}\right )^6}+\frac {x^3}{7 a \left (a+b \sqrt {x}\right )^7}\right )\) |
3.23.26.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(35)=70\).
Time = 3.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.30
method | result | size |
derivativedivides | \(\frac {10 a}{3 b^{6} \left (a +b \sqrt {x}\right )^{3}}-\frac {5 a^{4}}{3 b^{6} \left (a +b \sqrt {x}\right )^{6}}-\frac {1}{b^{6} \left (a +b \sqrt {x}\right )^{2}}+\frac {2 a^{5}}{7 b^{6} \left (a +b \sqrt {x}\right )^{7}}+\frac {4 a^{3}}{b^{6} \left (a +b \sqrt {x}\right )^{5}}-\frac {5 a^{2}}{b^{6} \left (a +b \sqrt {x}\right )^{4}}\) | \(99\) |
default | \(\frac {10 a}{3 b^{6} \left (a +b \sqrt {x}\right )^{3}}-\frac {5 a^{4}}{3 b^{6} \left (a +b \sqrt {x}\right )^{6}}-\frac {1}{b^{6} \left (a +b \sqrt {x}\right )^{2}}+\frac {2 a^{5}}{7 b^{6} \left (a +b \sqrt {x}\right )^{7}}+\frac {4 a^{3}}{b^{6} \left (a +b \sqrt {x}\right )^{5}}-\frac {5 a^{2}}{b^{6} \left (a +b \sqrt {x}\right )^{4}}\) | \(99\) |
trager | \(\frac {\left (-1+x \right ) \left (-a^{12} b^{8} x^{6}+7 a^{10} b^{10} x^{6}-21 a^{8} b^{12} x^{6}+42 a^{6} b^{14} x^{6}+105 a^{4} b^{16} x^{6}+231 a^{2} b^{18} x^{6}+21 b^{20} x^{6}+28 a^{14} b^{6} x^{5}-197 a^{12} b^{8} x^{5}+595 a^{10} b^{10} x^{5}-1050 a^{8} b^{12} x^{5}+42 a^{6} b^{14} x^{5}-1953 a^{4} b^{16} x^{5}+231 a^{2} b^{18} x^{5}+210 a^{16} b^{4} x^{4}-1442 a^{14} b^{6} x^{4}+4213 a^{12} b^{8} x^{4}-6608 a^{10} b^{10} x^{4}+9240 a^{8} b^{12} x^{4}+42 a^{6} b^{14} x^{4}+105 a^{4} b^{16} x^{4}+140 a^{18} b^{2} x^{3}-770 a^{16} b^{4} x^{3}+1498 a^{14} b^{6} x^{3}-932 a^{12} b^{8} x^{3}-6608 a^{10} b^{10} x^{3}-1050 a^{8} b^{12} x^{3}+42 a^{6} b^{14} x^{3}+7 a^{20} x^{2}+91 a^{18} b^{2} x^{2}-623 a^{16} b^{4} x^{2}+1498 a^{14} b^{6} x^{2}+4213 a^{12} b^{8} x^{2}+595 a^{10} b^{10} x^{2}-21 a^{8} b^{12} x^{2}+7 a^{20} x +91 a^{18} b^{2} x -770 a^{16} b^{4} x -1442 a^{14} b^{6} x -197 a^{12} b^{8} x +7 a^{10} b^{10} x +7 a^{20}+140 a^{18} b^{2}+210 a^{16} b^{4}+28 a^{14} b^{6}-b^{8} a^{12}\right )}{21 \left (-b^{2} x +a^{2}\right )^{7} \left (a^{14}-7 a^{12} b^{2}+21 a^{10} b^{4}-35 b^{6} a^{8}+35 b^{8} a^{6}-21 a^{4} b^{10}+7 a^{2} b^{12}-b^{14}\right )}-\frac {16 b a \,x^{\frac {7}{2}} \left (7 b^{4} x^{2}+14 a^{2} b^{2} x +3 a^{4}\right )}{21 \left (-b^{2} x +a^{2}\right )^{7}}\) | \(588\) |
10/3*a/b^6/(a+b*x^(1/2))^3-5/3/b^6*a^4/(a+b*x^(1/2))^6-1/b^6/(a+b*x^(1/2)) ^2+2/7*a^5/b^6/(a+b*x^(1/2))^7+4*a^3/b^6/(a+b*x^(1/2))^5-5/b^6*a^2/(a+b*x^ (1/2))^4
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (35) = 70\).
Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 4.37 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx=-\frac {21 \, b^{12} x^{6} + 231 \, a^{2} b^{10} x^{5} + 105 \, a^{4} b^{8} x^{4} + 42 \, a^{6} b^{6} x^{3} - 21 \, a^{8} b^{4} x^{2} + 7 \, a^{10} b^{2} x - a^{12} - 16 \, {\left (7 \, a b^{11} x^{5} + 14 \, a^{3} b^{9} x^{4} + 3 \, a^{5} b^{7} x^{3}\right )} \sqrt {x}}{21 \, {\left (b^{20} x^{7} - 7 \, a^{2} b^{18} x^{6} + 21 \, a^{4} b^{16} x^{5} - 35 \, a^{6} b^{14} x^{4} + 35 \, a^{8} b^{12} x^{3} - 21 \, a^{10} b^{10} x^{2} + 7 \, a^{12} b^{8} x - a^{14} b^{6}\right )}} \]
-1/21*(21*b^12*x^6 + 231*a^2*b^10*x^5 + 105*a^4*b^8*x^4 + 42*a^6*b^6*x^3 - 21*a^8*b^4*x^2 + 7*a^10*b^2*x - a^12 - 16*(7*a*b^11*x^5 + 14*a^3*b^9*x^4 + 3*a^5*b^7*x^3)*sqrt(x))/(b^20*x^7 - 7*a^2*b^18*x^6 + 21*a^4*b^16*x^5 - 3 5*a^6*b^14*x^4 + 35*a^8*b^12*x^3 - 21*a^10*b^10*x^2 + 7*a^12*b^8*x - a^14* b^6)
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (36) = 72\).
Time = 1.57 (sec) , antiderivative size = 619, normalized size of antiderivative = 14.40 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx=\begin {cases} - \frac {a^{5}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {7 a^{4} b \sqrt {x}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {21 a^{3} b^{2} x}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {35 a^{2} b^{3} x^{\frac {3}{2}}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {35 a b^{4} x^{2}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {21 b^{5} x^{\frac {5}{2}}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{8}} & \text {otherwise} \end {cases} \]
Piecewise((-a**5/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 7*a**4*b*sqrt(x)/(21*a**7*b**6 + 1 47*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3 *b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/ 2)) - 21*a**3*b**2*x/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8 *x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2 ) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 35*a**2*b**3*x**(3/2)/(21*a**7 *b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b** 13*x**(7/2)) - 35*a*b**4*x**2/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441* a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**1 1*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 21*b**5*x**(5/2)/(21* a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/ 2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21 *b**13*x**(7/2)), Ne(b, 0)), (x**3/(3*a**8), True))
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (35) = 70\).
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.28 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx=-\frac {1}{{\left (b \sqrt {x} + a\right )}^{2} b^{6}} + \frac {10 \, a}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{6}} - \frac {5 \, a^{2}}{{\left (b \sqrt {x} + a\right )}^{4} b^{6}} + \frac {4 \, a^{3}}{{\left (b \sqrt {x} + a\right )}^{5} b^{6}} - \frac {5 \, a^{4}}{3 \, {\left (b \sqrt {x} + a\right )}^{6} b^{6}} + \frac {2 \, a^{5}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{6}} \]
-1/((b*sqrt(x) + a)^2*b^6) + 10/3*a/((b*sqrt(x) + a)^3*b^6) - 5*a^2/((b*sq rt(x) + a)^4*b^6) + 4*a^3/((b*sqrt(x) + a)^5*b^6) - 5/3*a^4/((b*sqrt(x) + a)^6*b^6) + 2/7*a^5/((b*sqrt(x) + a)^7*b^6)
Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.49 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx=-\frac {21 \, b^{5} x^{\frac {5}{2}} + 35 \, a b^{4} x^{2} + 35 \, a^{2} b^{3} x^{\frac {3}{2}} + 21 \, a^{3} b^{2} x + 7 \, a^{4} b \sqrt {x} + a^{5}}{21 \, {\left (b \sqrt {x} + a\right )}^{7} b^{6}} \]
-1/21*(21*b^5*x^(5/2) + 35*a*b^4*x^2 + 35*a^2*b^3*x^(3/2) + 21*a^3*b^2*x + 7*a^4*b*sqrt(x) + a^5)/((b*sqrt(x) + a)^7*b^6)
Time = 6.00 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.02 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx=-\frac {\frac {a^5}{21\,b^6}+\frac {x^{5/2}}{b}+\frac {5\,a\,x^2}{3\,b^2}+\frac {a^3\,x}{b^4}+\frac {5\,a^2\,x^{3/2}}{3\,b^3}+\frac {a^4\,\sqrt {x}}{3\,b^5}}{a^7+b^7\,x^{7/2}+21\,a^5\,b^2\,x+7\,a\,b^6\,x^3+7\,a^6\,b\,\sqrt {x}+35\,a^3\,b^4\,x^2+35\,a^4\,b^3\,x^{3/2}+21\,a^2\,b^5\,x^{5/2}} \]